A small rock with mass m moves at velocity v0 and hits a rod of mass M as shown. The rod is initially at rest. The collision is completely inelastic.
a) What is the angular velocity of the rod after the collision?
b) What is the maximum angle the rod makes with the vertical?
To find the angular velocity of the rod after the collision, we need to apply the principle of conservation of linear momentum and angular momentum.
a) Conservation of linear momentum:
In an inelastic collision, the objects stick together after the collision and move as one. Therefore, the initial momentum of the rock and the rod combined (before the collision) is equal to their final momentum (after the collision).
Initial linear momentum = m * v0 (since the rock mass is m and its velocity is v0)
Final linear momentum = (m + M) * vf, where vf is the final velocity of the combined system (rock + rod)
Since the collision is completely inelastic, the final velocity of the system is zero (vf = 0).
(m + M) * 0 = m * v0
0 = m * v0
v0 = 0
Since the initial velocity of the rock is zero, the angular velocity of the rod after the collision is also zero.
b) To find the maximum angle the rod makes with the vertical, we need to analyze the energy of the system.
Before the collision, the rock has kinetic energy, given by (1/2) * m * v0^2.
After the collision, all the kinetic energy is converted into potential energy due to the rod's rotational motion.
When the rod reaches its maximum angle with the vertical, it momentarily comes to rest. At this point, all the kinetic energy is converted into potential energy.
Equating the kinetic energy before the collision with potential energy at the maximum angle:
(1/2) * m * v0^2 = (1/2) * I * ω^2 * h
Where I is the moment of inertia of the rod, ω is the angular velocity, and h is the height to which the center of mass of the rod is raised.
Since the angular velocity is zero (as found in part a), this equation becomes:
(1/2) * m * v0^2 = 0
This implies that the maximum angle the rod makes with the vertical is 0 degrees.